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Question: The height of mercury barometer is h when the atmospheric pressure is \(10^{5}Pa\), the pressure at ...

The height of mercury barometer is h when the atmospheric pressure is 105Pa10^{5}Pa, the pressure at x in the shown figure is

A. 105Pa10^{5}Pa
B. 0.8×105Pa0.8\times{10^{5}Pa}
C. 0.2×105Pa0.2\times{10^{5}Pa}
D. 120×105Pa120\times{10^{5}Pa}

Explanation

Solution

The above figure shows a setup of a static fluid. The horizontal surface of mercury will experience an exertion by the atmosphere. Pressure at point y can be calculated by formula of pressure. So by equating two equations for pressure at the horizontal surface we can easily encounter our answer.
As per the given data;
Height at which point x is situated =h−h5=h-\dfrac{h}{5}
Atmospheric pressure P∘=105P_\circ ={{10}^{5}} Pa

Formula used:
Pressure, P=P∘+ρghP=P_\circ +\rho gh

Complete answer:
Hydrostatic pressure refers to the pressure exerted by a fluid (gas or liquid) at any point in space within that fluid, assuming that the fluid is incompressible and at rest.
Pressure within a liquid depends only on the density of the liquid, the acceleration due to gravity, and the depth within the liquid. The pressure exerted by such a static liquid increases linearly with increasing depth.

Mathematically;
Pressure, P=P∘+ρghP=P_\circ +\rho gh
Where, P∘P_\circ the pressure due to empty space or atmospheric formula
So pressure at the bottom surface of the glass tube of the barometer (Py)(P_{y}) .will be;
(Assume that pressure due to the empty space in the glass tube is zero)
Py=0+ρghP_{y}= 0+\rho gh
Py=ρghP_{y}=\rho gh
Let us see the pressure at y due to point x under the glass tube:
Px=ρgh−ρgh5P_x=\rho gh-\rho g\dfrac{h}{5}
Px=ρgh−ρgh5P_x=\rho gh-\rho g\dfrac{h}{5}
=ρgh(1−15)=\rho gh\left ( 1-\dfrac{1}{5} \right )
We know that, In case of static fluid, pressure on the horizontal level is constant at every point.
Thus;
Py=ρgh=105PaP_{y}=\rho gh= 10^{5}Pa
By putting the value ofρgh\rho gh:
Px=105(1−15)P_x={{10}^{5}}\left( 1-\dfrac{1}{5} \right)
=105×45= 10^{5}\times \dfrac{4}{5}
Px=0.2×105P_x=0.2\times {{10}^{5}}

So, the correct answer is “Option C”.

Note:
Read the question carefully. Remember that we don’t have to consider the whole height of the glass tube. Most importantly this is a static fluid setup so apply properties according to that only.